An LQR Controller Design Approach For Pitch Axis ... · An LQR Controller Design Approach For Pitch Axis Stabilisation Of 3-DOF Helicopter System Mrs. M. Bharathi*Golden Kumar** Abstract

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1398 ISSN 2229-5518

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http://www.ijser.org

An LQR Controller Design Approach For

Pitch Axis Stabilisation Of 3-DOF

Helicopter System

Mrs. M. Bharathi*Golden Kumar**

Abstract

In this project an attempt has been made to adopt an LQR controller design approach for PITCH axis stabilization of 3DOF Helicopter System. The

presentation in this report is not limited to only design a controller for the stabilization of PITCH axis model of 3DOF Helicopter but at same time

it shows good performance. Some useful basic control systems concept related to Riccaati equation, controllability of the system and PID

controller have been also presented to understand the content of the project. The report first develops a transfer function and state space model

to represent the PITCH axis dynamics of 3DOF Helicopter system and then LQR controller design steps are explained in brief.

The investigated state feedback controller design technique is an optimal design method and it is directly applicable to unstable pitch axis model

of 3DOF Helicopter.

To show the effectiveness of the investigated method, the report also demonstrates the comparative studies between LQR and PID controllers. The

results of the closed loop system performance with LQR controller and PID controller separately are also shown.

1.Introduction

3DOF Helicopter System (shown in Fig. 1.1) is composed of the

base, leveraged balance, balancing blocks, propellers and some

other components. Balance posts to base as its fulcrum, and the

pitching. Propeller and the balance blocks were installed at the two

ends of a balance bar. The propeller rotational lift, turning a balance

bar around the fulcrum so pitching moves, using two propeller

speed difference, turning a balance bar along the fulcrum to do

rotational movement. Balance the two poles installed encoder, used

to measure the rotation axis, pitch axis angle, in the two propeller

connecting rod installed an encoder, which is used to measure

overturned axis angle. Two propellers using brushless DC motors,

provide the impetus for the propeller. By adjusting the balance rod

installed in the side of the balance blocks to reduce propeller motor

output. All electrical signals to and from the body are transmitted

via slip ring thus eliminating the possibility of tangled wires and

reducing the amount of friction and loading about the moving axes.

Preparation of the experimental guidance on the purpose is to tell

users how to design a controller, to control the helicopter in

accordance with the desired angle and speed of

movement.

Figure 1.1: 3 DOF Helicopter systems

The theory of optimal control is concerned with operating a

dynamic system at minimum cost. The case where the system

dynamics are described by a set of linear differential equation and

the cost is described by a quadratic functional called LQ problem,

one of the main results in the theory is that solution is provided by

the LQR, a feedback controller. First, we make a detail analysis and

modeling on 3DOF helicopter from its mechanism and features and

get its modeling motion equations by the knowledge of physics.

From the analysis of the model, the system is with the problem of


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non-linear and state interference. First, we get the linear state space

through linearity of the system, and then we use the theory of LQR

to get the optimal state feedback controller from the linear state

space.

1.2 Motivation

The motivation for doing this project was primarily an interest in

undertaking a challenging project in an interesting area of research.

I found the 3DOF Helicopter system as an appropriate area of

research of my interest, and using LQR controller design methods

for checking its controllability and robustness was my contribution

in this research paper. LQR controller is usually used in industry

especially in chemical process and aerospace industry. LQR problem

is one of the most fundamental and challenging control problems

and

inthis method; controller is very easy to design and also increases

the accuracy of state variable by estimating the state. It takes care

of the tedious work done by the control system engineers in

optimizing the controllers. However the engineer needs to specify

the weighting factor and compare the result with the specified

desired goals. This means that the controller synthesis is an iterative

process, where the engineer judges to produce optimal controllers

through simulation and computation and then adjusts the weighting

factor to get a controller more in line with the specified design

goals this computing and simulation work for controller synthesis,

motivated us to work on this project.

1.3 Objectives

Design and simulation of LQR controller for pitch axis stabilization

of 3 DOF helicopter system (using MATLAB).

2. Mathematical Modelling

It is composed of the base, leveraged balance, balancing blocks,

propellers and some other components. Balance posts to base as its

fulcrum, and the pitching. Propeller and the balance blocks were

installed at the two ends of a balance bar.

The propeller rotational lift, turning a balance bar around the

fulcrum so pitching moves, using two propeller speed difference,

turning a balance bar along the fulcrum to do rotational movement.

Balance the two poles installed encoder, used to measure the

rotation axis, pitch axis angle, in the two propeller connecting rod

installed an encoder, which is used to measure overturned axis

angle.

Two propellers, using brushless DC motors, provide the impetus for

the propeller. By adjusting the balance rod installed in the side of

the balance blocks to reduce propeller motor output. All electrical

signals to and from the body are transmitted via slip ring thus

eliminating the possibility of tangled wires and reducing the amount

of friction and loading about the moving axes.

Three differential equations to describe the dynamics of the system.

A simple set of differential equations is developed as follows:


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2.1 Pitch axis

Consider the diagram in Fig.2.1 Assuming the roll is zero, then

the pitching axis torque by two propeller motors lift the F1 and

F2. Therefore, the pitch propeller axis total lift Fh＝F1＋F2.When

the lift Fh is greater than the gravity G Helicopter rise. Instead

the helicopter dropped. Now, assuming zero roll, the differential

equation is:

Figure 2.1: Pitch axis dynamics

= ( ) (2.1)

= ( ) (2.2)

Where,

is the moment of inertia of the system about the pitch

axis, = +

.

is the mass of balance blocks.

is the total mass of two propeller motor.

and are voltages applied to the front and back motors resulting

in force and .

is the force constant of the motor /propeller combination.

is the distance from the pivot point to the propeller motor.

is the distance from the pivot point to the balance

blocks.Ignoring Tg in equation 3.2.2 we get

= (2.3)

Now, Taking Laplace transform of (2.2.3) we get:

( ) = ( )

( )

( ) =

Substituting the value of =12N/V, =0.88m, =1.8145kg. in the

above equation,we can get the transfer function of 3 DOF helicopter

system.

( )

( )=

This equation gives the pitch transfer function of 3 DOF.

2.2.State Space Modelling of Pitch axis for 3DOF:

We know that:

(2.10)

(2.11)

l1V1 c/Je + l1V2 c/ Je (2.12)

lpV1 c/Jp + l1V2 c/Jp (2.13)

l1p/Jt (2.14)

Assuming that :

(2.15)

(2.16)

Now we have to find A and B matrix for 3DOF Helicopter system

using the above seven linear differential equation:

[

]

[

]

So we have the linearized equation of above state-space A and B as:

[

]

=

[ ]

+ [

]


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3.Reserach Methodology

3.1 LQR controller design method

The LQR optimal control principle is, by system equations:

= AX + Bu

Determine matrix K that gives the optimal

Control vector: u (t) = -K*x(t)

Such that the performance index is minimized:

J = ∫ (

)

In which Q is positive definite (or semi -positive definite) hermitian

or real symmetric matrix R is positive definite hermitian or real

symmetric matrix.

Fig. 3.1: Optimal LQR controller diagram

The second term on the right of the equation is introduced in

concern of energy loss. Matrix Q and R determine the relative

importance of error and energy loss. Here, it is assumed that the

control vector u(t) is unbounded.

Weighting Matrices selection

One way of expressing the performance index mathematically is

through an objective function of this form:

J =∫ ∫

For Simplicity we assume Q and R as Diagonal matrix. Thus the

Objective J reduces to :

J =

Here, the Scalars q1...,qn, r....,rm can be looked upon as relative

weights between different performance terms in the objective J.

The key design problem in LQR is to translate performance

specifications in terms of the rise time, overshoot, bandwidth, etc.

into relative weights of the above form. There is no straightforward

way of doing this and it is usually done through an iterative

process either in simulations or on an experimental setup. Once

the matrices Q and R are completely specified, the controller gain

K is found by solving the Riccati equation.

[

] (3.1)

U1 and U2 are the maximum acceptable value of the input voltages.

And matrix Q can be found using Bryson’s rule:Q

=

[ ]

Where according to Bryson’s rule:

Q11 is1/ maximum acceptable value of (pitch angle) 2

Q22 is 1/ maximum acceptable value of (roll angle) 2

Q33 is 1/ maximum acceptable value derivative of (pitch angle) 2

Q44 is 1/ maximum acceptable value of derivative of (roll angle) 2

Q55 is 1/ maximum acceptable value of (travel rate) 2

Q66 is 1/ maximum acceptable value of (damping ratio) 2

Q77 is 1/ maximum acceptable value of (ϒ) 2

Substituting the above values:

Q =

[

]

3.2 PID controller design approach pitch axis of 3DOF Helicopter

system

The PID control scheme is named after its three correcting terms,

whose sum constitutes the manipulated variable. The proportional,

integral, and derivative terms are summed to calculate the output of

the PID controller.


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Figure 3.2: PID controller

As seen in Fig.2.2 the different terms associated with the controller

and its operations are being explained in detail below. The block

contains the three different parameters namely Proportional, Integral

and derivative. The final form of the PID algorithm is:

( ) ( ) ∫ ( )

( ) (3.0)

Where, =Proportional Gain, =Integral Gain,

=Derivative Gain, e =Error, t =Instantaneous time

Proportional term

The proportional term produces an output value that is proportional

to the current error value. The proportional response can be

adjusted by multiplying the error by a constant called the

proportional gain constant. A high proportional gain results in a

large change in the output for a given change in the error. If the

proportional gain is too high, the system can become unstable.

Integral Term

The contribution from the integral term is proportional to both the

magnitude of the error and the duration of the error. The integral in

a PID controller is the sum of the instantaneous error over time and

gives the accumulated offset that should have been corrected

previously. The accumulated error is then multiplied by the integral

gain and added to the controller output. The integral term

eliminates the residual steady-state error that occurs with a pure

proportional controller.

Derivative Term

The derivative of the process error is calculated by determining the

slope of the error over time and multiplying this rate of change by

the derivative gain. The magnitude of the contribution of the

derivative term to the overall control action is termed the derivative

gain. The derivative term slows the rate of change of the controller

output. Derivative control is used to reduce the magnitude of the

overshoot produced by the integral component and improve the

combined controller-process stability.

Pitch PID Controller

The Pitch axis model is given by equation (2.3):

Where, V1+V2 = Vs

We design the PID controller of the form as follows:

( ) ∫( ) (3.1)

ɛis the actual pitch angle, ɛ is the desired pitch angle

Now substituting the values we get:

( ) ∫( ) (3.2)

Taking Laplace transform, the closed loop Transfer function is given

by :

( ) ( ) ( ) ( )

( )

( )

( )= -

(3.3)

4.ResultAnalysis

4.1 Controllability of the system

A system is said to be controllable at time t, if it is possible by

means of an unconstrained control vector to transfer the system

from any initial state x(t) to any other state in a finite interval

of time. In fact, the conditions of controllability may govern the

existence of a complete solution to the control system design

problem. The solution to this problem may not exist if the system

considered is not controllable. Although most physical systems are


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controllable, corresponding mathematical models may not possess

the property of controllability. Then it is necessary to know the

conditions under which a system is controllable.

4.1.1 Complete State Controllability of Continuous-Time Systems:

Consider the continuous-time system.

= AX + Bu (4)

Where:

x = state vector (n-vector),u = control signal (scalar) ,

A = n X n matrix, B = n X 1 matrix

The system described by Equation (4.0) is said to be state

controllable at t = to if it is possible to construct an

unconstrained control signal that will transfer an initial state to

any final state in a finite time interval to t0≤t≤t1,. If every state is

controllable, then the system is said to be completely state

controllable.

We now derive the condition for complete state controllability.

Without loss of generality, we can assume that the final state is

the origin of the state space and that the initial time is zero.

The solution of equation (4.0) is:

X(t) = X(0) + ∫ ( ) ( )

(4.1)Applying the definition of

complete state controllability:

X( ) = 0 = X(0) + ∫ ( ) ( )

(4.2)

Or,We know that:

X(0) = ∫ ( )

(4.3)

∑ ( )

(4.4)

Substituting the equation (4.4) in (4.3) we get:

X(0)= ∑

∫ ( ) ( )

(4.5)

Let us put,∫ ( ) ( )

=

The equation (4.5) becomes:

X(0) = ∑

= [B AB B][

] (4.6)

If the system is completely state controllable, then, given any

initial state x(O),This requires that the rank of the n X n matrix

be ‘n’.

[B AB B]

From this analysis, we can state the condition for complete state

controllability as fol1ows:The system given by Equation (4.0) is

completely state controllable if and only if the vectors B, AB, . . .

. An-1Bare linearly independent, or the n X n matrix is of rank n.

[B AB B]

The result just obtained can be extended to the case where

the control vector u is r-dimensional. If the system is described by

= AX + Bu

Where u is an r-vector, then it can be proved that the condition for

complete state Controllability is that the n X nr matrix.

[B AB B]

B of rank n, or contain n linearly independent column vectors. The

matrix

[B AB B]

Is commonly called the Controllability matrix.

Controllability for Pitch axis dynamic model of 3 DOF Helicopter

system

We have a state space model of the helicopter system as follows:

A

[ ]

[

]

Using MATLAB the controllability matrix of the system is obtained,

M=


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[

]

The rank of the above matrix M is 7 which is equal to order of the

system matrix A[MATLAB].

Therefore the system is controllable.

4.2 Open loop response for pitch axis

The differential equation for pitch axis dynamics from equation (2.3)

is given by:

= ( )

Ignoring Tg in the above equation we get:

=

Now, Taking Laplace transform of we get:

( ) = ( )

( )

( ) =

Substituting the value of =12N/V, =0.88m, =1.8145kg. in the

above equation,we can get the transfer function of 3 DOF helicopter

system. Finally the open loop transfer function is:

( )

( )=

Figure 4.1: Open loop response

The Pitch axis model of Helicopter system is unstable as it gives

unbounded output for the bounded input signal. It is shown in the

figure 4.1

4.3 State feedback controllerof Pitch axis model for Helicopter

system:

The plant state space model is already explained in section 2.2.4

and it follows that

[

]

0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Step Response

Time (seconds)

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plit

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[

]

The weighting matrices Q & R are selected based on the theory

which is explained in the section 3.1. The Matrices Q and R are

finally chosen by the equations 3.1 and 3.2.

[

]

[

]

Now using the MATLAB command:[K S E] =LQR (A, B, Q/1000, and

R), the state feedback gain matrix and closed loop pole of the

system are given by the matrices K and E respectively:

[

]

Here, all the closed loop poles (Eigen values) of the system are

either lying in the left half of the s-plane or on the imaginary axis,

therefore our designed system is stable.

Natural frequency and damping ratio of the closed loop system is

also found using MATLAB code:[Wn,Z,P]=damp(A-B*K)and we

found that response for the 2nd order system for each value of

natural frequency (Wn) and damping factor (Z) are acceptable. It is

shown in the following figure.

Figure 4.2: Response for diff. Wn and Z

4.4 Pitch PID Controller using the values of state feedback gain K

The state feedback gain matrix K is given by [Section 4.3]

[

]

And we can also write the above Matrix K as:

[

]

And full state feedback results in a controller those feedback two

voltages:

[

] [

]

[

]

( ) (4.7)

Now comparing the result with equation (3.1):

( ) ∫( )

Now comparing the above two equations, the gains we obtain from

LQR design can still be used for pitch controller as follows:

4.4.1 Simulation results

Using equation 3.3 the closed loop transfer function of the system is

given by:

( )

( )= -

Now, substituting the values Kc=12, l1=0.88, Je= 1.8145, Kep= -

3.7192, Ked= -1.2168 and Kei= -1.4906, we got:

( )

( )=

The above transfer function is obtained using extracted values Kep,

Kei and Ked from designed state feedback gain matrix K as

explained in section 4.3. 0 2 4 6 8 10 12 14 16 18

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plit

ude


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The response of the above closed loop transfer function is obtained

and it is shown in figure 4.3.

Figure 4.3: Designed closed loop system response

Now, substituting the values Kc=12, l1=0.88, Je= 1.8145,

Kep= -2.0852, Ked= -0.8698 and Kei= -0.2, we get:

( )

( )=

The response of the above closed loop transfer function is obtained

and it is shown in figure 4.4.

Figure4.4: Closed loop system response for reference PID value

4.4.2 Real time system response

The real time simulation is done using Helicopter PID control

diagram [6]

Figure 4.5:3DOF Helicopter MATLAB Real Time Control Diagram

Double click the “Pitch PID” block to set pitch PID parameters.

Figure 4.6: Pitch PID block diagram

Double click the “Kp” block to set proportional parameter of pitch

PID as the simulation results, and double click “OK” to save

parameters.

Figure 4.7: Kp Block

Double click the “Ki” block to set integral parameter of pitch PID as

the simulation results, and double click “OK” to save parameters

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plit

ude

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plit

ude


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Figure 4.8: Ki Block

Double click the “Kd” block to set derivative parameter of pitch PID

as the simulation results, and double click “OK” to save parameters.

Figure 4.9:Kd Block

Case 1: System response for 35 degrees (reference) pitch angle

Figure 4.10: Tracking for 35 degree

Result: We found that system is stable and tracking the reference

input succesfully.

Case 2: System response for 45 degrees:

Figure 4.11: Tracking for 45 degrees

Result: In this case also system is stable and tracking the input

signal.

Case 3: System response for 55 degrees:-

Figure 4.12: Tracking for 55 degrees

Result: The system is stable and tracking the reference signal.

4.6 Comparative studies between LQR and PID controllers

The PID controller parameters Kp, Kd and Ki have been found using

LQR state feedback gain matrix K [section 4.3] and then closed loop

system performance analysed. The following time domain

performance parameters are obtained.


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Figure 4.13: Designed closed loop response

Rise Time: 0.4291s,Settling Time: 4.8948sOvershoot: 14.6289%,Peak:

1.1463,Peak Time: 1.0470s

Again we have taken PITCH PID controller values and closed loop

system response obtained which is shown in the figure

Figure 4.14: Closed loop response of reference PID values

Rise Time:0.6021s,Settling Time:7.9956s,Overshoot: 7.6368%Peak:

1.0764,Peak Time: 1.3321s

Conclusion:

We observe that the system is giving better performance for the

designed controller with respect to existing reference PID controller

but performance is deteriorated only in terms of overshoot.

5.CONCLUSION AND FUTURE SCOPE OF WORK

5.1 Conclusion

In this project an optimal design approach has been chosen to

design a state feedback controller for PITCH axis model of 3DOF

Helicopter system. First we developed the Mathematical model of

3DOF Helicopter system and then stability & performance of the

modelled system is carried out.

The theory of LQR controller design has been investigated and a

different approach based on Bryson rule has been also adopted to

select the weighting matrices which are used in controller synthesis.

The selected project is also demonstrated successfully in real time

platform and it is followed by the comparison with existing design.

Simulation analysis is also shown in the report.

5.2 Future Work:

In this project an LQR controller is synthesized for PITCH axis

stabilization for 3 DOF helicopter systems and the same approach

can be extended for Travel and Roll axes for the same system.

*Proffessor and Head, Department of Electronics

And Instrumentation, Bharath University

** Dept. of E&I, Bharath University,

Chennai – 600072,

INDIA

6.REFERENCES

[1] Li Qing, Yin Xiuyun, Liu Zhengao and Fanfan,

“Construction of helicopter loop simulation system and

model identification”, ICEMI 2007

[2] Pei-Ran Li, Tao Shen, “The research of 3DOF Helicopter

tracking control”, August 2007

[3] Mariya A.Ishutkina, “Design and Implementation of

supervisory safety controller for 3DOF Helicopter system”,

MIT, June 2004

[4] Bo Zheng and Yisheng Zhong, “Robust attitude regulation

of 3DOF helicopter benchmark”, IEEEfeb 2011.

[5] David G. Hill, “Optimal control theory for application”,

Springer, 2004

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plit

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0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (seconds)

Am

plit

ude


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[6] Technical Manual-3DOF Helicopter Systems, GOOGOL

TECH, 2011

[7] Robert W. Newcomb, “Optimal control theory”, 1970

[8] Katsuhiko Ogata, Modern Control engineering”, 2010

[9] Peter Whittle, “Optimal Control –Basic & Beyond, John

Wiley & Sons, 1996

[10] Frank L. Lewis, “Applied Optimal Control &Estimation,

PHI, 1988

Documents

An LQR Controller Design Approach For Pitch Axis ... · An LQR Controller Design Approach For Pitch Axis Stabilisation Of 3-DOF Helicopter System Mrs. M. Bharathi*Golden Kumar** Abstract